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96 4. Families of Elliptic Curves and Geometric Properties of Torsion Points

(5.1) Invariants of the Curve E[a, b]. For the curve E[a, b]definedby

3
2
2
y = x + ax + bx
the following hold:

2 5 3
c 4 = 16 a − 3b , c 6 = 2 9ab − 2a ,
2
c 3 a − 3b 3
4 2 2 4 8
= 2 b a − 4b , j = = 2 .
2
b 2 a − 4b
The two basic special cases are:
3
(1) j = 12 if and only if a = 0 where we abbreviate the notation E[0, b] = E[b]:
2
3
y = x + bx.
2
2
(2) j = 0ifand onlyif3b = a = (3c) for characteristic unequal to 3. This is
2 2 3 2 2 2 2
the curve E[3c, 3c ] where y = x + 3cx + 3cx x,or E[3c, 3c ]: y =
3
2
3
3
3
(x + c) − c . It has the form y = x − c after translation of x by c.
Before giving the formulas for the isogeny, we observe that the function h(a, b) =
2 2 2 2
(−2a, a − 4b) when iterated twice is h (a, b) = (4a, 16b) since (−2a) − 4(a −
2
2
4b) = 16b. This means h : k → k is a bijection for char(k) = 2, and it is its own
inverse up to the division of the coordinates by a power of 2, namely
2
h −1 (a, b) = (−a/2,(a + 4b)/16).
2
2
Finally, note that a − 4b is the discriminant of the quadratic x + ax + b.
(5.2) Formulas for the 2-Isogeny. The 2-isogeny with kernel {0,(0, 0)} is given by
2
ϕ : E[a, b] → E[−2a, a − 4b] where
2 2
y y x − b b b
ϕ(x, y) = , = x + a + , y 1 − .
x 2 x 2 x x 2
2
The formula for the dual isogeny ˆϕ : E[−2a, a − 4b] → E[a, b]isgiven by
2
y y 2 2
ˆ ϕ(x, y) = , x − a − 4b
4x 2 8x 2
2 2
1 a − 4b y a − 4b
= x − 2a + , 1 − .
4 x 8 x 2
Note that ˆϕ is given by the same formulas as ϕ up to powers of 2 in the coordi-
2
nates which reflects the fact that the dual ˆϕ maps E[−2a, a − 4b]to E[a, b] while
6
2
ϕ maps E[−2a, a − 4b]to E[4a, 16b]. Multiplying the equation for E[a, b]by2 ,
we see that (x, y) → (4x, 8y) is an isomorphism of E[a, b] → E[4a, 16b]. Hence
any property that holds for ϕ will also hold for the dual example an easy calculation

shows that ϕ(x, y) = (x , y ) satisfies